Vertex Ramsey properties of randomly perturbed graphs

Given graphs $F,H$ and $G$, we say that $G$ is $(F,H)_v$-Ramsey if every red/blue vertex colouring of $G$ contains a red copy of $F$ or a blue copy of $H$. Results of {\L}uczak, Ruci\'nski and Voigt and, subsequently, Kreuter determine the threshold for the property that the random graph $G(n,p)$ is $(F,H)_v$-Ramsey. In this paper we consider the sister problem in the setting of randomly perturbed graphs. In particular, we determine how many random edges one needs to add to a dense graph to ensure that with high probability the resulting graph is $(F,H)_v$-Ramsey for all pairs $(F,H)$ that involve at least one clique.


Introduction
For r P N, a sequence of (not necessarily distinct) graphs H 1 , . . . , H r , and a graph G, we say that G is pH 1 , . . . , H r q v -Ramsey if for every r-colouring of the vertices of G, there is some i P rrs for which G contains a copy of H i whose vertices are all coloured in the ith colour. Similarly, we say G is pH 1 , . . . , H r q-Ramsey if for every r-colouring of the edges of G, there is some i P rrs for which G contains a copy of H i whose edges are all coloured in the ith colour. In the case when r " 2 we take the convention that the colours used are red and blue. If H 1 "¨¨¨" H r " H we write e.g. pH 1 , . . . , H r q v -Ramsey as pH, rq v -Ramsey.
The classical question in Ramsey theory is to establish the smallest n P N such that the complete graph K n on n vertices is pH 1 , . . . , H r q-Ramsey. In general, by Ramsey's theorem such an n is known to exist though relatively few such Ramsey numbers are known precisely. In contrast to this, the analogous question in the setting of vertex colourings is completely trivial. Indeed, the pigeonhole principle implies K n is pH 1 , . . . , H r q v -Ramsey if n " vpH 1 q`¨¨¨`vpH r q´r`1 but not pH 1 , . . . , H r q v -Ramsey if n is any smaller. density to w.h.p. ensure the resulting graph is pK r , K s q-Ramsey for all values of pr, sq (except for the case when r " 4 and s ě 5). See [7] for other results on this topic.
In this paper, we focus on vertex Ramsey properties of randomly perturbed graphs; in particular we resolve the pH, K r q v -Ramsey problem for r ě 2 and arbitrary H. To state our results we first introduce the following notation. Definition 1.5. Fix some d P r0, 1s. Then for a pair of graphs pF, Hq, we say that p " ppnq is a perturbed vertex Ramsey threshold function for the pair pF, Hq at density d if: (i) For any qpnq " ωpppnqq and any sequence pG n q nPN of graphs of density 1 at least d with v Gn " n for each n P N, with high probability G n Y Gpn, qq is pF, Hq v -Ramsey. (ii) There exists a sequence of n-vertex graphs pG n q nPN of density at least d, such that if qpnq " opppnqq, then with high probability G n Y Gpn, qq is not pF, Hq v -Ramsey. We denote by ppn; F, H, dq, the 2 perturbed vertex Ramsey threshold function for pF, Hq at density d. If there exist C, c ą 0 such that qpnq ě Cppnq suffices for (i) and qpnq ď cppnq suffices for (ii), we say that the threshold function is sharp. If it is the case that every sufficiently large graph of density at least d is pF, Hq v -Ramsey then we define ppn; F, H, dq :" 0.
Note we can analogously define the perturbed vertex Ramsey threshold function for the r-coloured case; that is, given graphs H 1 , . . . , H r we define the threshold ppn; H 1 , . . . , H r , dq in the natural way. If H 1 "¨¨¨" H r " H we write ppn; H 1 , . . . , H r , dq as ppn; H, r, dq. Example 1.6. Casting Theorem 1.4 into this notation, we have that ppn; F, H, 0q " n´1 {m K pF,Hq , for a pair of graphs F, H with m 1 pF q ď m 1 pHq (when EpF q is nonempty and H is not a matching) and this threshold is sharp.
Before we state our main result it is instructive to consider the following result of Krivelevich, Sudakov and Tetali [19], which determines how many random edges need to be added to a dense graph to force the appearance of H as a subgraph. In our vertex-Ramsey framework, this corresponds to making the graph pK 1 , Hq v -Ramsey. The corresponding threshold probability requires the following definition. Definition 1.7. For a graph H, the appearance threshold for H in the random graph Gpn, pq is determined by the parameter Now, given any k P N, let mpH; kq :" min where the minimum is over all partitions of H into k induced subgraphs. 3 Suppose that G is a graph of density more than 1´1{pk´1q and we wish to find a copy of H in G Y Gpn, pq. Informally, we partition H into k parts H 1 , . . . , H k that are as sparse as possible, with the idea being to use the (few) edges of Gpn, pq to build the parts H i , and then find the edges between parts in the dense graph, thereby completing a copy of H. Note that mpH; kq " 0 if and 1 Here we refer to the standard density of G. That is dpGq " e G p v G 2 q . 2 As is the case in random graph theory, the threshold function is not uniquely determined but rather determined up to constants. 3 By a partition of H into k induced subgraphs, we mean there are k (possible empty) graphs H1, . . . , H k such that each Hi is an induced subgraph of H; the Hi are all pairwise vertex-disjoint; V pHq " V pH1q Y¨¨¨Y V pH k q.
only if χpHq ď k, in which case we can partition H into k independent sets. Then we do not require any random edges; the dense graph itself will already contain H.
Our main result (Theorem 1.11 below) essentially resolves the pH 1 , H 2 q v -Ramsey problem for randomly perturbed graphs for all pairs pH 1 , H 2 q involving at least one clique. To state this result we define some notation capturing the probabilistic vertex Ramsey thresholds for all pairs of graphs. Definition 1.9. Given graphs F and H, we write βpF, Hq :" That is, βpF, Hq is defined so that n´1 {βpF,Hq is the threshold for the property that Gpn, pq is pF, Hq v -Ramsey. We can now define our perturbed vertex Ramsey threshold, which is an extension of Definition 1.7. Definition 1.10. Given r P N, k ě 2 and a graph H, define m˚pK r , H; kq :" max Here the first maximum is taken over all tuples pr 1 , . . . , r k q of non-negative integers that sum to at most r´1; the minimum is over all partitions of H into k induced subgraphs; the final maximum is over all i such that H i contains at least one vertex.
Theorem 1.11. Let r P N and k ě 2 and H be a graph. Given any d ą 0 so that 1´1{pk´1q ă d ă 1´1{k, we have ppn; K r , H, dq " n´1 {m˚pKr,H;kq .
Note that Theorem 1.11 is general in the sense that it covers the full range of densities d P p0, 1q (not just small values of d). Further, Theorem 1.8 is precisely the r " 1 case of Theorem 1.11.
Although Theorem 1.11 does not always guarantee sharp thresholds, by analysing its proof one can see that, in all cases except when s P tt{2, tu, the threshold in Corollary 1.12 is determined by Theorem 1.4, which does provide a sharp result. Hence, we obtain the moreover part of the corollary. On the other hand, when s P tt{2, tu, the threshold probability in the corollary comes from the appearance of a subgraph, which does not have a sharp threshold.
Recall that in the random graph setting one needs Θpn 2´1{m K pKs,Ktq q random edges to ensure Gpn, pq is pK s , K t q v -Ramsey. Corollary 1.12 demonstrates that one needs far fewer random edges to make any dense n-vertex graph pK s , K t q v -Ramsey. However, the precise number of random edges depends (in a rather subtle way) on arithmetic properties of the pair ps, tq.
1.3. Some intuition for vertex Ramsey problems in randomly perturbed graphs. In this section our aim is to convince the reader that the vertex Ramsey problem for randomly perturbed graphs is in general more subtle than its counterpart in the random graph setting.
In Theorem 1.2 the threshold is universal in the following sense: the threshold for Gpn, pq being pH, rq v -Ramsey is the point above which every linear sized subset of Gpn, pq w.h.p. contains a copy of H. It is easy to see that this property guarantees a graph is pH, rq v -Ramsey (as one of the colour classes in any vertex r-colouring will have linear size). Thus, crucially the 'reason' for the location of the threshold is the same for every graph H (that is not a matching). Moreover, this reason is independent of the number of colours used.
Similarly, the threshold in Theorem 1.4 is universal. Indeed, given any sequence of graphs H 1 , . . . , H r as in the theorem, the intuition behind the threshold for the property of Gpn, pq being pH 1 , . . . , H r q v -Ramsey is the same: the threshold is the point at which the expected number of vertex-disjoint copies of H r is roughly the same order of magnitude as the maximal order of a H r´1 -free subgraph of Gpn, pq. (See the discussion in [17].) Again this threshold does not depend on the number r of colours.
On the other hand, the threshold for the perturbed vertex Ramsey problem can depend on the number of colours. Indeed, we saw in Corollary 1.12 that for every t ě 3 and d P p0, 1{2q the number of random edges required to ensure an n-vertex graph of density d is w.h.p. pK t , 2q v -Ramsey is significantly smaller than ppn; K t , 2, 0q; that is, significantly smaller than the number of edges needed to ensure Gpn, pq is w.h.p. pK t , 2q v -Ramsey. On the other hand, given any r ě 4, we actually have that ppn; K t , r, dq " ppn; K t , r, 0q for all d P p0, 1{2s. In fact, this phenomenon is part of a more general observation. Observation 1.13. Let r ě 4 and H be graph that is not a matching. Let d P p0, 1{2s. Then ppn; H, r, dq " n´1 {m 1 pHq " ppn; H, r, 0q.
Indeed, if p ě Cn´1 {m 1 pHq for some constant C, Theorem 1.2 shows that Gpn, pq itself will be pH, rq v -Ramsey w.h.p., and hence this is an upper bound on the perturbed vertex Ramsey threshold.
For the lower bound, take G to be a complete balanced bipartite n-vertex graph with vertex classes A and B. By Theorem 1.2, if p ď cn´1 {m 1 pHq for some constant c, then with high probability both Gpn, pqrAs and Gpn, pqrBs are not pH, 2q v -Ramsey. This therefore implies there exists a 4-colouring of the vertices of G Y Gpn, pq without a monochromatic copy of H.
One can view the threshold in Theorem 1.11 as universal, in the sense that it is governed by a single parameter, m˚pK r , H; kq (for all graphs H and r ě 3). However, the intuition behind where the parameter m˚pK r , H; kq comes from is more involved than the corresponding intuition discussed above for Theorems 1.2 and 1.4; we discuss this more when proving Theorem 1.11. Further, when proving Theorem 1.11 we were unable to establish how one can extend the statement to all pairs of graphs pH 1 , H 2 q (i.e. not just those where at least one of the H i is a clique). In general we expect the location of the vertex Ramsey threshold to depend heavily on subtle properties of the graphs under consideration and in this way a universal reason, if it even exists, will be challenging to come by.
1.4. Notation. Throughout the paper we omit floors and ceilings whenever this does not affect the argument. Further, we use standard graph theory and asymptotic notation. In particular, for a graph G, vpGq denotes the number of vertices of G and epGq the number of edges of G; note that we often condense this notation to v G and e G respectively. We say a graph is nonempty if it has a nonempty vertex set, and unless otherwise specified, we shall take the vertex set to be rv G s :" t1, 2, . . . , v G u. Given a hypergraph H and a set I Ă V pHq, deg H pIq denotes the number of edges of H that contain the set I.
Given a set X and r P N we write`X r˘f or the set of all subsets of X of size r. Similarly, if V is a set of vertices and F is a graph, we denote by`V F˘t he set of all possible copies of F supported on vertices in V . Here we consider these copies of F to be distinct if they have distinct sets of edges, where autpF q is the number of automorphisms of F .

1.5.
Organisation of the paper. In Section 2 we prove Theorem 1.11, handling the 0-statement in Section 2.1 and the 1-statement in Section 2.2. In the latter section, we shall require a 'robust' version of the 1-statement of Theorem 1.4 (Theorem 2.8), which we prove in Section 3. Finally, in Section 4 we give some concluding remarks.

Proof of the perturbed threshold
In this section we prove Theorem 1.11. We present the arguments for the 0-and 1-statements in separate subsections below.
2.1. The 0-statement. Here we will show the existence of a graph G of density at least d such that, when p " o´n´1 {m˚pKr,H;kq¯, with high probability the vertices of G Y Gpn, pq can be two-coloured without a red copy of K r or a blue copy of H.

2.1.1.
Kreuter's Theorem for families. In showing the existence of a good colouring, we will use the 0-statement of Theorem 1.4, which shows that Gpn, pq can be vertex-coloured while avoiding monochromatic subgraphs. However, in our application, we will have to avoid several subgraphs in the same colour class, and therefore need the following generalisation of Theorem 1.4 to families of graphs. If p " o`n´1 {m˘, then with high probability there is a red/blue-colouring of the vertices of Gpn, pq without a red copy of any graph F P F and without a blue copy of any H P H.
We remark that Proposition 2.1 is tight, since if p " ω`n´1 {m˘, then with high probability Gpn, pq is pF, Hq v -Ramsey by Theorem 1.4, where pF, Hq is the minimising pair in the definition of m.
The proof of Proposition 2.1 is nearly identical to the proof of the 0-statement of Theorem 1.4; that is, the case when both F and H each contain a single graph. We therefore simply sketch the key idea here and refer the reader to [17] for the details.
First we handle the degenerate cases. Suppose one of the families, say F, contains a graph F with no edges. The parameter βpF, Hq is then the appearance threshold for the graph H in Gpn, pq, and so if p " o`n´1 {m˘, we have that with high probability Gpn, pq has no copy of any graph from H, and so we can colour all its vertices blue. The other degenerate case is when all graphs in F and H have edges, but both families contain matchings, say F and H. In this case, βpF, Hq " 1. If p " o`n´1˘, then with high probability Gpn, pq is bipartite. We can thus two-colour its vertices such that each colour class is an independent set, and thus has no copy of any graph from F or H.
We may therefore assume that every graph in F has edges, and that H does not contain a matching, bringing us to the setting of Theorem 1.4. The proof now follows a similar scheme to other 0-statement proofs in random Ramsey settings, e.g. [15]. One begins by supposing for a contradiction that you cannot red/blue-colour the graph Gpn, pq avoiding red copies of graphs in F and blue copies of graphs in H. Using this fact one can define some set G of graphs obtained by 'gluing together' copies of graphs in F and in H in certain ways, and show that Gpn, pq must contain a graph in G. In [17] G is defined by way of an algorithm that finds a copy of some G P G in Gpn, pq. In order to do this they pass to a 'critical' subgraph G 1 of Gpn, pq which is minimal (in terms of copies of graphs in F and H) with respect to the property of not being able to 2-colour G 1 avoiding red copies of graphs in F and blue copies of graphs in H. In G 1 , one can see that for every copy T of a graph in F or H and every vertex v of T , there is a copy of a graph in the other family which intersects T exactly at the vertex v [17, Claim 1]. The algorithm [17, Procedure Hypertree] which builds a subgraph J of G 1 is then defined by repeatedly adding copies of some F P F or H P H, so that the copy intersects the previous copy in exactly one vertex. The proof then works by analysing this procedure and the graphs in G that can be found using this procedure. In particular, one keeps track of a function f piq which controls the exponent of the expected number of J i , where J i is the graph found in G 1 after i steps of the algorithm. The procedure will stop if the f piq gets too small or if the procedure continues for roughly log n steps. This will lead to a contradiction, as the graphs in G which are the possible outcomes of this procedure are all dense graphs and are either large or satisfy a very strong density condition [17,Claim 5] and hence are very unlikely to occur in Gpn, pq at this density. One can also bound the size of G [17, Claim 6] so that a union bound will guarantee that with high probability no such graph in G is found in Gpn, pq. In the calculations involved in the analysis [17, Claims 2,3 and 4] of the effect on f in each step of the algorithm, there are some minor changes in our setting as we have to consider the possibility of any member of our family being added by the algorithm to update the J i . However, it can be seen that adding a 'denser' graph than the graph which is in the minimising pair for the definition of m, will only help the situation, in that the function f can only decrease further, meaning that the resulting J i is at most as likely to appear in Gpn, pq as the J i obtained by adding the minimising graph as in the calculations in [17].

2.1.2.
Proof of the 0-statement. We may assume that m˚pK r , H; kq ą 0, as otherwise there is no 0-statement to prove. We take the dense graph G n to be the balanced complete k-partite graph on n vertices, which has density at least 1´1 k , and let V 1 , V 2 , . . . , V k be the k vertex classes.
Let pr 1 , . . . , r k q be the maximising vector in Definition 1.10. For each i P rks, we define a family of nonempty subgraphs of H by Proof. Suppose H R H j for some j P rks, and consider the partition which contradicts the definition of m˚pK r , H; kq.
In particular, each of the families H i is nonempty. We can now describe, for each i P rks, our colouring of the vertices in V i . Let F " tK r i`1 u and H " H i . By definition of H i , we have min F PF ,H 1 PH i βpF, H 1 q ě m˚pK r , H; kq. Since p " o´n´1 {m˚pKr,H;kq¯, it follows from Proposition 2.1 that with high probability we can colour V i such that Gpn, pqrV i s has neither a red K r i`1 nor a blue graph from H i . The following claim shows this gives a valid colouring of G n Y Gpn, pq, completing the proof of the 0-statement. Claim 2.3. With this colouring, G n Y Gpn, pq has neither a red K r nor a blue H.
Proof. By construction, the largest red clique in V i is of order at most r i . The largest red clique in V pG n Y Gpn, pqq " Y i V i therefore has at most ř i r i ď r´1 vertices, and hence the colouring is red-K r -free.
Suppose there was a blue copy of H, and let H " H 1 Y. . .YH k be the partition of H induced by the parts V i . By definition of m˚pK r , H; kq, there is some part i P rks with βpK r i`1 , H i q ě m˚pK r , H; kq (and H i ‰ H). It then follows that H i P H i , but our colouring of Gpn, pqrV i s avoids blue copies of any graph in H i , contradicting the existence of this blue copy of H.

The 1-statement.
To prove the 1-statement of Theorem 1.11, we need to show that whenever p " ω´n´1 {m˚pKr,H;kq¯, G n Y Gpn, pq will with high probability be pK r , Hq v -Ramsey. When G n is the complete k-partite graph, as it was in the proof of the 0-statement, this amounts to finding the sparse parts of the graphs in Gpn, pqrV i s, which can then be joined together since we have a complete k-partite graph.
However, in our more general setting, G n is an arbitrary graph of density d ą 1´1{pk´1q. By employing Szemerédi's Regularity Lemma [25], we shall find some structure in G n that mimics the behaviour of a complete k-partite graph. These structural results, together with probabilistic tools concerning the random graph Gpn, pq, are collected in the following subsections, before being used in the proof of Theorem 1.11 in Section 2.2.3.
2.2.1. Structure in dense graphs. Our application of the Regularity Lemma follows the standard lines. We present here the necessary definitions and properties of regular pairs, referring the reader to the survey of Komlós and Simonovits [16] for further details.
Definition 2.4. Given ε ą 0, a graph G and two disjoint vertex sets A, B Ă V pGq, the pair pA, Bq is ε-regular if for every X Ď A and Y Ď B with |X| ą ε|A| and |Y | ą ε|B|, we have |dpX, Y q´dpA, Bq| ă ε, where dpS, T q :" epS, T q{p|S||T |q for any vertex sets S and T .
In essence, the edges between a regular pair 'look random', in the sense that they are very well distributed. The next lemma showcases some beneficial properties of these regular pairs: small sets of vertices typically have many common neighbours, and subsets of regular pairs inherit a large degree of regularity. We omit the proofs of these facts, which can be found in [16].
Lemma 2.5. Let pA, Bq be an ε-regular pair in a graph G with dpA, Bq " d.
Szemerédi's Regularity Lemma then famously asserts that the vertices of any sufficiently large graph can be partitioned into a large but bounded number of parts, such that almost all pairs of parts are ε-regular. We shall not require the full strength of the Regularity Lemma, but only the following corollary, which follows in combination with Turán's Theorem [26].

2.2.2.
Probabilistic tools. While Proposition 2.6 gives us the desired structure in the dense graph, we also require a couple of results about the random graph. The first of these counts the number of copies of a fixed graph H in Gpn, pq. Following [10], we define the following parameter for H and p " ppnq, The lemma below shows that we are very unlikely to have significantly fewer copies of H than expected.
Lemma 2.7 (Janson's inequality). Let H be a nonempty graph, p " ppnq and H Ď`r ns H˘b e some family of Ωpn v H q potential copies of H on rns. Letting X be the random variable that counts the number copies of H in H which appear in Gpn, pq, we have that The proof of this lemma follows almost immediately from the main result of [11] (see also [10,Theorem 2.14]). Indeed, for each potential copy S P H of H, let X S be the indicator random variable for the event that S appears in Gpn, pq. Then X " ř SPH X S and [10, Theorem 2.14] implies that Second, we shall make use of the vertex Ramsey properties of Gpn, pq. We will need the following 'robust' version of the 1-statement of Theorem 1.4, which guarantees that we can find monochromatic copies of the desired graph that are suitably well-located. Furthermore, it shows that these Ramsey properties hold with sufficiently high probability to be applied several times. Theorem 2.8. Let F and H be graphs with 0 ă m 1 pF q ď m 1 pHq. Then there exist δ 0 , c ą 0 such that for all 0 ă δ ă δ 0 , t " tpnq ď exppn c q and η 0 ą 0, there exists a C ą 0 such that the following holds. Suppose that Then, if p ě Cn´1 {m K pF,Hq , the following holds with high probability in Gpn, pq. For any two-colouring of rns and every i P rts, there is either a red copy S P F i of F or a blue copy T P H i of H.
Notice that, unlike in Theorem 1.4, Theorem 2.8 allows for both F and H to be matchings. The proof is similar to that of the 1-statement of Theorem 1.4, but this strengthened version requires a few additional ideas and some careful analysis of the failure probabilities at each step. We defer these details until Section 3.2, and instead complete the proof of Theorem 1.11 next.

2.2.3.
Proof of the 1-statement. We first sketch the ideas behind the proof. By Proposition 2.6, we can find a k-tuple of vertex sets V 1 , V 2 , . . . , V k such that each pair is ε-regular and reasonably dense. We then hope to use the Ramsey properties of Gpn, pqrV i s to find suitable monochromatic subgraphs that can be pieced together to form a red K r or a blue H.
However, a naïve application of Theorem 1.4 will not work. Indeed, by definition of m˚pK r , H; kq, there is some vector pr 1 , . . . , r k q with ř i r i ď r´1 and some partition H " We can therefore expect that, for each i, we either find a red K r i`1 or a blue H i in any vertex colouring of Gpn, pqrV i s.
If these monochromatic subgraphs were all of the same colour, then we could hope to combine them to form a red clique (which would in fact be of size r`k´1, significantly larger than required) or a blue copy of H. However, we could well find red cliques in some parts and blue subgraphs in others, which would leave us unable to complete either of the desired graphs.
Instead, we must use the full power of Definition 1.10, which provides a suitable partition H " H 1 Y . . . Y H k not just for some vector pr 1 , . . . , r k q, but rather for all vectors pr 1 , . . . , r k q satisfying ř i r i ď r´1. We shall therefore proceed in stages, incrementally either increasing the size of a red clique or finding the next piece needed for a blue copy of H. We let r i denote the size of the largest red clique we have found in V i thus far, starting with r " 0.
Given the current vector r, we let H " H 1 Y . . . Y H k be the corresponding minimising partition of H. We then go through the parts in turn, applying the pK r i`1 , H i q v -Ramsey property of Gpn, pqrV i s to find a blue H i or a red K r i`1 . In the former case, we proceed to the next part. If we make it through each of the k parts, we will have found all the parts H i needed to build a blue copy of H.
Otherwise, in the latter case, we have increased the size of our red clique. We then update the vector r and the corresponding partition of H, return to the first part V 1 , and resume the process. Since this increases the size of our red clique, we will have built a red K r if this latter case occurs r times.
There are still technicalities that need to be dealt with -for instance, to ensure we can combine the monochromatic structures we find, we will need to restrict ourselves to the common neighbourhoods of the parts we have already found. This further requires us to only consider subgraphs with "many" common neighbours in all other parts, which is why we need the more robust 1-statement of Theorem 2.8. In the remainder of this section, we describe this algorithm in detail and show that it successfully returns one of the desired monochromatic subgraphs.
Given α ą 0 and p " ωpn´1 {m˚pKr,H;kq q, our goal is to show that for any n-vertex graph G n of density d ě 1´1{pk´1q`2α, the graph G n Y Gpn, pq is with high probability pK r , Hq v -Ramsey. Applying Proposition 2.6 to G n with some suitably small regularity parameter ε, 4 gives k pairwisedisjoint vertex sets V 1 , V 2 , . . . , V k , such that each pair pV i , V j q is ε-regular of density at least α.
At several stages in the algorithm, we will, for some i, find a (constant sized) subgraph Γ Ă Gpn, pqrV i s, and will then want to shrink all the other parts V j to the common neighbours in G n of the vertices of Γ. We shall therefore call Γ popular if its vertices have at least p α 2 q v Γ |V j | common G n -neighbours in each V j , j ‰ i. Lemma 2.5 ensures that most potential copies of Γ will be popular, 4 For our purposes, it suffices to take ε " krv H pr`2 v H q , where δ 1 is the minimum value of δ0 from Theorem 2.8 when the graph F is a clique on at most r vertices and the graph H in the theorem is a subgraph of our given graph H. and that when we shrink the sets V j to their large common neighbourhoods, the pairs will remain ε 1 -regular of density α 1 , for some larger ε 1 and some slightly smaller α 1 . By choosing the initial value of ε small enough, we can ensure that all subsequent values of ε 1 remain small, while the densities α 1 are always at least α 2 . We first find copies of the subgraphs of K r and H that are likely to appear in Gpn, pq. Let t :" maxts P rrs : mpK s q ď m˚pK r , H; kqu and let G :" tHrU s : U Ď V pHq, mpHrU sq ď m˚pK r , H; kqu. We then define the graph Γ to be the disjoint union of v H copies of K t together with one copy of each graph in G. Then, for each i P rks in turn, find a popular copy Γ i of Γ in Gpn, pqrV i s, and shrink all other parts V j to the common neighbours of V pΓ i q in V j . If Γ is an independent set, clearly one can find these popular copies. Otherwise, noting that mpΓq ď m˚pK r , H; kq, Lemma 2.7 ensures that we find these popular copies with high probability. Note that, at the end of this process, for all i the graph Γ i remains in the set V i .
We can now start the procedure sketched earlier. We shall denote by R i the largest red clique found in Gpn, pqrV i s thus far, initially setting R i " H for all i P rks. The vector r will be defined by r i :" v R i , so we begin with r " 0.
The outer loop of the algorithm runs as long as ř i r i ď r´1, which means we have not yet found a red K r . In this case, we take the minimising partition H " H 1 Y . . . Y H k for the vector r in Definition 1.10, and try to find a blue copy of H according to this partition.
The inner loop of the algorithm runs over i P rks. If H i " H, then there is nothing to find in Gpn, pqrV i s, and so we proceed to the next part. Otherwise, we shall show that Gpn, pqrV i s is robustly pK r i`1 , H i q v -Ramsey; that is, we will find a popular blue H i or a popular red K r i`1 . If we have a blue H i , we let B i be this copy of H i , shrink all other parts V j to the common neighbours of V pB i q, and then proceed to the next part.
On the other hand, if we find a red K r i`1 , then we have increased the size of our red clique. We then set R i to be this larger clique and shrink all other parts V j to the common neighbours of V pR i q. We update the vector r, replacing r i with r i`1 , and then break the inner loop and proceed to the next iteration of the outer loop (trying to find the new optimal partition of H, starting in V 1 ).
Since we shrink to common neighbourhoods at each step, we ensure that the pieces we find in Gpn, pqrV i s can be combined to form the graphs we need in G n Y Gpn, pq. In particular, if the inner loop were to run through all k steps, then Y i B i would give a blue copy of H. On the other hand, each iteration of the outer loop increases the size of our red clique, and after r iterations Y i R i would give a red K r . Thus, after finitely many steps, the algorithm returns either a blue H or a red K r , showing that G n Y Gpn, pq is indeed pK r , Hq v -Ramsey.
To complete the proof, we need to show that Gpn, pqrV i s will always be robustly pK r i`1 , H i q v -Ramsey, which we will mostly achieve through use of Theorem 2.8. However, this only applies when r i , epH i q ě 1. For the degenerate cases, we will need to make use of the graphs Γ i we found at the beginning. Suppose first that r i " 0. By definition, we have m˚pK r , H; kq ě βpK r i`1 , H i q " βpK 1 , H i q " mpH i q, and so H i appears in Γ i . Then either this copy of H i is completely blue, or we find a red K 1 , and so Γ i Ď Gpn, pqrV i s is indeed pK 1 , H i q v -Ramsey. The other case, when epH i q " 0, follows similarly. This time we have m˚pK r , H; kq ě βpK r i`1 , H i q " mpK r i`1 q, and so Γ i contains v H copies of K r i`1 . Either one of them is completely red, in which case we are done, or we have v H blue vertices, which in particular gives a blue copy of H i . This leaves us with the case when both K r i`1 and H i have edges. Again, by definition, we have m˚pK r , H; kq ě βpK r i`1 , H i q. Thus, since p " ωpn´1 {m˚pKr,H;kq q, Theorem 1.4 shows we should expect Gpn, pqrV i s to be pK r i`1 , H i q v -Ramsey. However, we need this to be true for all sets V i that could arise, and also need to find popular monochromatic copies of K r i`1 or H i , and thus we apply Theorem 2.8 instead.
Note that the sets V i that arise are the common neighbourhoods of a bounded number of vertices, and hence there are only polynomially many possibilities. Moreover, as these are always neighbourhoods of popular subgraphs, there is some constant η 0 ą 0 such that |V i | ě η 0 n.
For each such set V i , we define a triple pU j , F j , H j q P U , where we take U j :" V i , we let F j be all possible popular copies of K r i`1 in V i , and let H j be all possible popular copies of H i in V i . Lemma 2.5 (and our choice of small ε) ensures thatˇˇ`U j We therefore satisfy all the requirements of Theorem 2.8, and can conclude that with high probability, whenever we require Gpn, pqrV i s to be pK r i`1 , H i q v -Ramsey, it will be. As there are only finitely many pairs pK r i`1 , H i q to consider, it follows that the algorithm succeeds with high probability overall, completing the proof.

Robust Ramsey properties of random graphs
The aim of this section is to give a proof of Theorem 2.8. Although our proof here is similar to that of Kreuter [17], we choose to give the details as the argument is somewhat delicate and our proof departs from the original in some key steps. In particular, instead of using Turán's theorem to estimate the maximal size of a set of vertex-disjoint copies of a given graph (as done by Kreuter [17]), we use a probabilistic approach (as in [1, Lemma 7.3.1]) which allows us to analyse the relevant subgraph counts at every step of the proof and guarantee that we find monochromatic copies of the graphs on the desired vertex sets. We first give some probabilistic tools and intermediate lemmas before embarking on the proof.
Lemma 3.1. Suppose tA i : i P Iu is a finite set of events in some probability space and for each i P I, let X i be the indicator random variable for the event A i . Write i " j if the events A i and A j are not independent. Further, let X :" ř iPI X i be the sum of the indicator random variables and define ∆ : where the sum is over all ordered pairs pi, jq (including diagonal terms). Then for all t ą 0, 3.1.2. Janson's inequality for a refined random graph. Given all the copies of a fixed H in Gpn, pq, it will be useful for us to look at a random subset of the copies where each copy is selected with probability q independently of all other choices. Formally, let ρpH; qq :`r ns H˘Ñ t0, 1u be a function that randomly assigns 1 with probability q and 0 with probability 1´q to each copy of H in K n , independently of the other choices. Then let Gpn, pq H q be the random graph obtained by revealing Gpn, pq and ρpH; qq and taking all the copies of H on rns such that all the edges of the copy appear in Gpn, pq and the copy was assigned a 1 by ρpH; qq. Lemma 3.2. Let H be a graph with at least one edge, p " ppnq, q " qpnq, and H Ď`r ns H˘b e some family of Ωpn v H q potential copies of H on rns. Letting X q be the random variable that counts the number copies of H in H which appear in Gpn, pq H q on rns, we have that PrX q ď ErX q s{2s ď expp´ΩpΦ H,p qq`expp´Ωpqn v H p e H qq.
The lemma follows from an application of Lemma 2.7, which gives concentration for the number of copies S P H of H that appear in Gpn, pq. Each copy is then kept with probability q, independently of the others, and so Chernoff's inequality (see e.g. [10, Theorem 2.1]) gives concentration for the number of these copies that appear in Gpn, pq H q . 3.1.
3. An exponential upper tail bound. Janson's inequality (Lemma 2.7) allows us to conclude that the probability that the number of embeddings of a graph H in Gpn, pq is significantly smaller than its expectation is exponentially small. On the other hand, one can use Lemma 3.1 to give a bound on the probability that the number of copies of H is much higher than expected. However, the concentration given by Lemma 3.1 is not enough for our purposes. We therefore need the following bound for the upper tail of the distribution of subgraph counts in random graphs. This is a simplification of the main result in [12] and the proof is almost identical. The only departing point from the exposition in [12] is that Lemma 3. Then there exists some c " cpH, ǫq ą 0 such that the following holds. Let H Ă`r ns H˘b e some family of ǫn v H potential copies of H. Letting X be the random variable that counts the number copies S P H which appear in Gpn, pq on rns, we have that PrX ě 2ErXss ď exp´´cΦ 1 e H¯.

3.1.4.
Kim-Vu polynomial concentration. The last tool we need is the result of Kim and Vu [14] (see also [1,Section 7.8]). We state here a simplified version which is catered to our purposes. Lemma 3.4. Given k P N, let c :" 8´1p4k!q´1 {p2kq , and let H " pV, Eq be a k-uniform hypergraph with |V | " N , |E| " M . Now consider the set V 1 obtained by keeping each vertex of V with some probability q " qpN q P r0, 1s, independently of the other vertices. We are interested in the random variable Y :" e HrV 1 s and we fix µ :" ErY s " M q k . Then setting 2k˙.

3.2.
Proof of Theorem 2.8. Towards proving Theorem 2.8, we first prove some lemmas. For a fixed nonempty graph H, we define to be the set of graphs that can be obtained by taking the union of two distinct copies of H which intersect in at least one vertex. Recall also the definition of Gpn, pq H q from Section 3.1.2.
Lemma 3.5. Let H be a graph with at least one edge, and let Φ 1 :" mintn, Φ H,p u. Then there exists C " CpHq ą 0 such that the following holds for all q " qpn, pq such that q ď Φ 1 {pn v H p e H q. Let X F be the random variable that counts the number of copies of a graph F in Gpn, pq H q . Then we have that Proof. This is a simple application of Chebyshev's inequality, Lemma 3.1. The proof of [10, Theorem 3.29] contains a similar calculation.
Let us fix someH " H 1 Y H 2 P HpHq and show that XH ď C 1 ErX H s 2 {Φ 1 with probability at least 1´CΦ 1 {ErX H s 2 for some C 1 ,C ą 0. The conclusion will then follow by a union bound, as there are finitely many possibleH P HpHq. First, let us upper bound the expectation of XH as follows. Defining J " H 1 X H 2 as the intersection of the two copies of H that compriseH, we have that for some appropriately defined C 1 ą 0, using that v J ě 1 and n v J p e J ě Φ H,p if e J ‰ 0. We now turn to concentration and look to apply Lemma 3.1. In order to do this, we need an upper bound estimate on ∆ which counts the expected number of non-independent pairs of copies ofH in Gpn, pq H q . That is, it counts the number of pairs of copies ofH which overlap in at least one edge. So let us fix some graph H˚" in at least an edge. There are finitely many such H˚and our upper bound on ∆ will come from summing over all such possible intersecting pairs of copies ofH. Letx be 2 if H˚" H 1 Y H 2 " H 1 1 Y H 1 2 is a single copy ofH, 1 if H i " H 1 j in Hf or some i, j P t1, 2u and 0 otherwise. In other words,x indicates the number of 'repeated' copies of H in H˚. Swapping the indices 1 and 2 if necessary, let J 1 " H 1 X H 2 , J 2 " H 1 1 X pH 1 Y H 2 q and J 3 " H 1 2 X pH 1 Y H 2 Y H 1 1 q such that each J i contains at least one vertex. This is possible due to the fact that H 1 and H 2 intersect in at least a vertex inH and the two copies ofH intersect in at least an edge. Then we have that for appropriately defined C 2 ą 0, using that at leastx of J 2 and J 3 are copies of H, and using that ErX H s ď qn v H p e H ď Φ 1 in the final step (recall that by hypothesis q ď Φ 1 {pn v H p e H q). Thus, summing over all possible H˚, we get that ∆ ďCErX H s 2 {Φ 1 for someC ą 0 and by Lemma 3.1, forC " 4C{C 12 . Thus summing over allH P HpHq and taking a union bound on the failure probabilities, we can choose C ą 0 appropriately so that the statement of the lemma is satisfied.
Let Ů k H denote the graph obtained by taking k vertex-disjoint copies of H. We say that a set K P`r ns k˘i s a transversal of a copy S of Ů k H if K contains one vertex from each of the copies of H that comprise S. Further, given K Ď`r ns k˘, we say a copy of Ů k H on rns is K-spanning if it contains a set from K as a transversal.
Lemma 3.6. Let H be a graph with at least one edge, k P N and δ ą 0. Set δ 1 :" δpkv H q k . Then there exists a c ą 0 and n 0 P N such that if p " ppnq ě n´1 {mpHq and q " qpn, pq satisfy qn v H p e H ą plog nq 3k then the following holds for all n ě n 0 . Suppose that K Ă`r ns k˘i s such that |K| ď δn k . Letting Y be the random variable that counts the number of K-spanning copies of Ů k H in Gpn, pq H q , we have that Proof. It suffices to prove the lemma in the case when |K| " δn k . For this, we split the analysis of Gpn, pq H q into looking at the random edges given by Gpn, pq and the random function ρpH; qq : rns H˘Ñ t0, 1u separately. Firstly consider the K-spanning copies of Ů k H in the complete graph K n . There are at most δ 1 n kv H such copies and each appears with probability p ke H . Thus, in expectation, the number of K-spanning copies of Ů k H in Gpn, pq is at most δ 1 pn v H p e H q k . Moreover, Lemma 3.3 tells us that the count of such copies in Gpn, pq is at most twice this with probability at least 1´exp´´c 3.3 Φ 1 ke H¯, where c 3.3 " cpH, δq as given by Lemma 3.3 and Φ " Φ´Ů k H, p¯. Now note that where we split subgraphs J Ď Ů k H according to their subgraphs J i in the i th copy of H in Ů k H and in the second step we single out a j " jpJq such that J j Ď J has a nonempty edge set.
Applying Lemma 3.3 also to the counts of Ů k 1 H, for smaller values of k 1 , we can conclude that there exists a c 1 ą 0 so that, with probability at least 1´exp´´c 1 ΦpH, pq 1 ke H¯, there are at most 2δ 1 pn v H p e H q k K-spanning copies of Ů k H in Gpn, pq and there are at most 2pn v H p e H q k 1 copies of Ů k 1 H in Gpn, pq for all 1 ď k 1 ď k´1. On the other hand there are at least δn kv H {p2kv H q! K-spanning copies of Ů k H in K n . So by Lemma 2.7 there is a c 2 ą 0 such that with probability at least 1´exp`´c 2 ΦpH, pq˘, there are at least 2δ 2 pn v H p e H q k K-spanning copies of Ů k H in Gpn, pq where δ 2 :" δ{4p2kv H q!. Now we condition on all these events occurring in Gpn, pq and turn to analyse the effect of ρpH, qq. We know that each K-spanning copy in Gpn, pq appears in Gpn, pq H q with probability q k and we will obtain concentration via a simple application of Lemma 3.4. Indeed, consider the auxiliary k-uniform hypergraph H whose vertex set is given by copies of H in Gpn, pq and whose edge set is given by copies of H which comprise a K-spanning copy of Ů k H in Gpn, pq. From above we have that H has at most 2n v H p e H vertices (the copies of H in Gpn, pq) and between 2δ 2 pn v H p e H q k and 2δ 1 pn v H p e H q k edges. We also know, from the concentration on the number of copies of Ů k 1 H in Gpn, pq, that for any set I of i copies of H with 1 ď i ď k, the number of edges of H containing I is at most 2pn v H p e H q k´i . Thus, Lemma 3.4 tells us that conditioning on the outcome of Gpn, pq as above, with probability at least the number of K-spanning copies of Ů k H in Gpn, pq H q is at most 4δ 1 pn v H p e H qq k , where c 3.4 is the constant given by Lemma 3.4. The conclusion then follows from a simple calculation on the error probability that either the counts in Gpn, pq are not as desired or the count in Gpn, pq H q is too high, given that we get the desired counts in Gpn, pq.
We now turn to proving Theorem 2.8.
Proof of Theorem 2.8. It suffices to prove the theorem in the case when p " Cn´1 {m K pF,Hq for some sufficiently large C ą 0. We begin with a calculation. Let The upper bound on ℓ follows because Now we turn to the proof of the theorem. We first fix constants. We fix δ 0 ą 0 such that and c ą 0 such that c ă c 1 4v F e F e H . Further, for each 0 ď i ď t we fix η i :" |U i |{n so that η i ě η 0 for all i. Further, fix p1{pv H !q´δq 2 pη 0 η i q v H 16C 3.5 for all 0 ď i ď t, where C 3.5 " C 3.5 pHq is the constant obtained from Lemma 3.5. By considering a large enough constant C, we expose Gpn, pq in two rounds so that Gpn, pq " G 1 pn, p 1 q Y G 2 pn, p 2 q, with p 1 , p 2 ě C 1 n´1 {m K pF,Hq for C 1 such that C 1 ą 2 log v H γ v F 0 . Let us briefly sketch the proof which splits into proving two main claims. The first claim states that with high probability in G 1 , for each i P rts, there is a (large) subfamily D i Ă H i of pairwise vertex-disjoint copies of H, all of whose edges appear in G 1 . We define (3.5) W i :" tW Ă U i : |W X T | " 1 for all T P D i u to be the sets which can be obtained by choosing one vertex from each copy of H in D i . The second claim is that with high probability in G 2 , for each i and each set W P W i , there is a copy of F which lies in`W F˘X F i whose edges appear in G 2 . The proof then follows easily from these two claims. Indeed, consider a red/blue colouring of Gpn, pq " G 1 Y G 2 and some i P rts. If there is no blue copy of H in H i then in particular, every copy of H in D i must contain a red vertex. By choosing one red vertex in each copy T of H in D i , we get a set W P W i which is entirely red. The second claim then tells us that this set hosts a copy of F which lies in F i and so we are done. It remains to prove the two claims above. In order to prove the first claim, it will be useful to consider the refined random graph G 1 pn, p 1 q H q introduced in Section 3.1.2. As G 1 pn, p 1 q H q is a subgraph of G 1 pn, p 1 q it will suffice to find our family D i of copies of H in G 1 pn, p 1 q H q . So we fix Now we apply Lemma 3.5, observing that Φ 1 ě n ℓ due to our calculation at the beginning of this proof (Φ 1 " Φ H,p ě n ℓ if ℓ ă 1 and Φ 1 " n ℓ " n if ℓ " 1). As the expected number of H in G 1 pn, p 1 q H q is Ωpqn v H p e H q " Ωpn ℓ q, we have that with high probability (with probability at least 1´Opn´ℓq), in G 1 pn, p 1 q H q there are at most C 3.5 q 2 n 2v H p 2e H n ℓ " γ 0 n ℓ overlapping copies of H. For a given i P rts, we can conclude from Lemma 3.2 that with high probability there at least p1{pv H !q´δqq|U i | v H p e H 2 ě 2γ i n ℓ copies of H in H i which lie in G 1 pn, p 1 q H q . As this holds with probability at least 1´expp´n c 1 q, we have that this holds for all i P rts with high probability. Thus we obtain a family D i Ă H i of vertex-disjoint copies of H which appear in G 1 by taking the copies in H i that appear in G 1 pn, p 1 q H q and deleting one copy from any pair of overlapping copies. Our calculations above guarantee that with high probability, for every i P rts, D i has size at leastñ i :" γ i n ℓ and we restrict each family to one of size exactlyñ i .
We now turn to the second exposure, namely G 2 " G 2 pn, p 2 q, and look to prove that for every i P rts and every set W P W i there is a copy of F in`W F˘X F i which appears in G 2 , where W i is as defined in (3.5). Fixing an i P rts and a W P W i , we consider G 2 restricted to W . We look to apply Lemma 2.7 and so need a lower bound on the parameter which is calculated with respect to the vertex set W . As at the beginning of the proof, we set J Ă H to be the minimising subgraph in the definition of ℓ (3.1) and use that for all I Ď F with e I ą 0, we have that We conclude thatΦ i ěΦ 0 ě γ v F 0 C 1 n ℓ . As t ď exppn c q and for each i, and |W i | " pv H qñ i ď exppn ℓ log v H q, we can take a union bound and conclude from Lemma 2.7 that for all choices of i P rts and W P W i , we have that there are at leastñ v F i p e F 2 copies of F on W in G 2 with high probability. Note here that we used that . It remains to prove that for each i P rts and W one of these copies of F belongs to F i .
To this end we define B i :"`U i F˘z F i to be the copies of F which do not lie in our desired collection. We will upper bound the number of copies S of F in B i which appear in G 2 , such that each vertex of S lies in a different copy of H in D i . In order to do this, we return to analyse our construction of D i and in particular our use of Gpn, pq H q . Let K i be the collection of v F -sets in U i which host a copy of F in B i and note that |K i | ď δ|U i | v F . We say a set K P K i is dangerous if each vertex of K is contained in a distinct copy of H in D i . In order to be dangerous, a set K has to lie in a transversal of a copy of Ů v F H in Gpn, pq H q restricted to U i (see the paragraph before Lemma 3.6 for the relevant definitions). Therefore in order to upper bound the number of dangerous sets, it suffices the upper bound the number of K i -spanning copies of Ů v F H in Gpn, pq H q rU i s. It follows then from Lemma 3.6 that for all i P rts, there are at most dangerous sets with high probability, using that δ ă δ 0 . As for a fixed i P rts, this holds with probability 1´expˆ´Ωˆn ℓ 2v F e H˙˙a nd t ď exppn c q, we can conclude that there at mostñ v F i {p8v F !q dangerous sets for each i P rts with high probability.
Finally, we calculate how many copies of F in G 2 are hosted on dangerous sets. For a fixed i, we consider G 2 restricted to the vertex set D i :" Y T PD i V pT q. We have that |D i | " v Hñi and from the previous paragraph we may assume that there are at mostñ v F i {8 potential copies of F on dangerous sets in D i . Each of these appears with probability p e F and by Lemma 3.3 we have that with probability at least 1´expˆ´Ωˆn ℓ e F˙˙, there are at mostñ v F i p e F {4 copies of F in G 2 which are hosted on dangerous sets. The failure probability here follows from a calculation of the appropriate Φ F,p 2 similar to the calculation ofΦ above. Thus we can take a union bound to conclude that for all i P rts, there are at mostñ v F i p e F {4 copies of F P B i which lie in D i , whose edges appear in G 2 and whose vertices are contained in distinct copies of H in D i . Thus with high probability, for all i P rts and for all W P W i , there is a copy T P`W F˘X F i of F , whose edges appear in G 2 , as required.

Concluding remarks
In this paper, we have determined, at essentially every density d, the perturbed vertex Ramsey threshold ppn; K r , H, dq for cliques versus arbitrary graphs. One could investigate how these thresholds change with the introduction of additional colours, but the most pressing problem that remains open is to extend our results to all pairs of graphs pF, Hq, with the symmetric case F " H of particular interest. Our methods do provide lower and upper bounds on the threshold in the general case, which we discuss below.
We start with the 1-statement, where we wish to know what p ensures G n Y Gpn, pq is pF, Hq v -Ramsey when G n is a graph of density more than 1´1{pk´1q. Recall that in our algorithmic proof of the 1-statement in Theorem 1.11, we worked in an ε-regular k-tuple in G n , using the vertex Ramsey properties of the random graph in each part to iteratively grow a red clique or try to build a copy of H.
In the general setting, when we seek a red copy of F instead, we can adopt the same approach. The main difference is that there are many ways we could try to build F over the k parts. To keep track of these, we define a partial partition of F to be a partition of the vertices V pF q " where W ‰ H. This represents the stage in the algorithm where we have found red subgraphs F rU i s in the parts V i , and W represents the vertices of F that are still missing. Thus, when we try to extend this red subgraph, we will require Gpn, pqrV i s to be pF rU i Y tu i us, H i q v -Ramsey for some optimal choice of u i P W and partition H " H 1 Y . . . Y H k . In this way, we either get one vertex closer to having a red copy of F , or we find one of the parts we need for a blue copy of H. Note that when F " K r , all that matters is the size |U i | and not the set U i itself, since each induced subgraph of K r is itself a clique. Hence we recover the bound of Theorem 1.11. Unfortunately, this upper bound need not be tight. For instance, when F and H are complete bipartite graphs, it is not hard to see that m˚pF, H; 4q " 0. However, by considering sets U i that, for each i, span both colour classes of F , we can ensure that each subgraph F rU i s has edges, which results in the right-hand side of (4.1) being positive.
One issue with (4.1) is that it considers all partial partitions of F , but we need only maximise over those that could feasibly arise in the algorithm. While it may not be easy to describe these partitions explicitly, we can construct the family of feasible partial partitions recursively.
To do so formally, we define the extension function f which, given the sets U 1 , . . . , U k of a partial partition of F , returns the vertices pu 1 , . . . , u k q P W k that are used to extend the red subgraph. For each such extension function, we can build the family Fpf q of feasible partitions in the following way. We start with pH, . . . , Hq P Fpf q. Then, for each i P rks, we add pU 1 , . . . , U i´1 , U i Y tf p U q i u, U i`1 , . . . , U k q to Fpf q, provided this is still a partial (and not complete) partition of F . Note that this represents the larger red subgraph we would obtain if, in Gpn, pqrV i s, we would find a monochromatic red subgraph when applying the pF rU i Y tf p U q i us, H i q v -Ramsey property.
We then need only maximise over the feasible partitions Fpf q, and can choose the extension function f that gives the lowest possible threshold. That is, we have the upper bound Note again that in the case F " K r , the choice of extension function f is irrelevant, since all that matters are the sizes |U i |. This is strictly better than (4.1), as one can find an extension function f that shows m˚pF, H; 4q " 0 whenever F and H are bipartite. Unfortunately, even (4.2) need not be tight, as we should also have m˚pF, H; 3q " 0 for such F and H, but the right-hand size is positive when we only have three parts. It would therefore be very interesting to find a sharper bound for the 1-statement. A useful step in that direction could be to characterise which extension functions f are optimal for a given graph F .
In the other direction, we can provide lower bounds on m˚pF, H; kq by generalising the colouring we gave in proving the 0-statement of Theorem 1.11. We shall once again take G n to be a complete k-partite graph, and will describe how one can colour the vertices of the random graphs Gpn, pqrV i s to avoid both a red F and a blue H in G n Y Gpn, pq.
To this end, we call a k-tuple pF 1 , . . . , F k q of families of nonempty graphs a k-cover of F if, for any k-partition V pF q " U 1 Y . . . Y U k of the vertices of F , there is some i P rks such that F rU i s P F i . That is, a k-cover is a collection of induced subgraphs that are bound to appear in any k-partition of F .
Given this definition, we have the following lower bound. When F " K r , this recovers the bound from Theorem 1.11, since we have k-covers of the form F i " tK r i`1 , . . . , K r u, where ř i r i " r´1. To describe the colouring in the general case, fix a maximising k-cover pF 1 , . . . , F k q, let β˚be the right-hand side of (4.3), and let p " opn´1 {β˚q . For each i P rks, we define H i " tH 1 Ď H : @F 1 P F i , βpF 1 , H 1 q ě β˚u. As before, one can argue that H P H i , and so these families are all nonempty. Applying Proposition 2.1, we can colour the vertices of Gpn, pqrV i s so as to avoid any red graph from F i and any blue graph from H i .
It is now tautological that this colouring of G n YGpn, pq has neither a red F nor a blue H. Suppose for contradiction there is a red copy of F , partitioned as F " F 1 Y . . . Y F k . Since pF 1 , . . . , F k q is a k-cover of F , there is some i with F i P F i , but then there is no red F i in Gpn, pqrV i s. On the other hand, if there is a blue H, partitioned as H " H 1 Y . . . Y H k , then we must have some i P rks such that H i ‰ H and βpF 1 , H i q ě β˚for all F 1 P F i . But then H i P H i , and so there is no blue H i in Gpn, pqrV i s either.
The challenge arises from the fact that when F is not a clique, there could be many ways to partition it into k induced subgraphs, and so there will be a wide variety of complicated k-covers.